Why the Derivative of -sin(x) is cos(x) | Understanding the Chain Rule for Calculating Derivatives

-sin(x) is the derivative of?

The derivative of -sin(x) is cos(x)

The derivative of -sin(x) is cos(x).

To understand why, let’s start by defining the derivative of a function. The derivative of a function f(x) measures its rate of change at any given point. It tells us how the function behaves as x changes slightly.

In this case, we are looking at the function -sin(x). The negative sign in front simply indicates that the function is reflected about the x-axis.

Now, to find the derivative of -sin(x), we can use the chain rule. The chain rule states that if we have a composite function, y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x).

To apply the chain rule, we let f(u) = -sin(u) and g(x) = x. Then, our composite function becomes y = f(g(x)) = -sin(g(x)) = -sin(x).

Now, let’s differentiate the composite function using the chain rule:
dy/dx = f'(g(x)) * g'(x)

First, we find the derivative of -sin(u) with respect to u. The derivative of sin(u) is cos(u). Taking the negative sign into account gives us f'(g(x)) = -cos(g(x)).

Next, we find the derivative of g(x) = x, which is simply 1. Thus, g'(x) = 1.

Now, we can substitute these derivatives back into the chain rule equation:
dy/dx = -cos(g(x)) * 1 = -cos(x).

Therefore, the derivative of -sin(x) is -cos(x).

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